Quantum transport equations for highly disordered systems
with spin-orbit interaction
G. J. Morgan1, B. Spisak2, and A. Paja2
1
Department of Physics and Astronomy, University of Leeds,
Leeds LS2 9JT, UK
2
Department of Solid State Physics, Faculty of Physics and Nuclear Techniques,
University of Mining and Metallurgy, 30-059 Krakow, Poland
2
Abstract
Disordered metallic systems exhibit much higher resistivity than polycrystalline
metals because structural disorder causes weak localisation of electrons, which in
turn diminishes conductivity. This localisation comes from quantum interference
and cannot be described in terms of the usual Boltzmann equation. On the other
hand quantum interference is very sensitive to various factors, which can destroy
it, for example spin-orbit scattering. In this work we present the quantum transport
equations, which take into account the spin of conduction electrons and the
interaction with the scattering potential.
3
Introduction
Amorphous materials are very important for many technological applications.
Their
physical properties can be described in terms of the theory of disordered systems. The oneelectron Hamiltonian for such systems reads
2 2
H 
  V r  rn 
2m
n
(1)
where the second term represents random static potential and includes all possible types of
scattering. We cannot use the Bloch functions to describe electron states because the system has
no translational invariance.
Nevertheless we mark the electron states with wave vector k
assuming that electron behaves as a plane wave at least between two successive collisions. This
is essential for developing the theory of electron transport in such media.
Theoretical models of electron transport
First working models:
 for liquid metals – Ziman (1961) [1]
 for liquid metallic alloys – Faber & Ziman (1965) [2]
 for amorphous alloys – Meisel & Cote (1977) [3]
 for highly resistive alloys – Morgan, Howson, & Saub (1985) [4]
In highly disordered media the scattered particle has non-zero probability of return to the initial
position after several elastic collisions. Then quantum interference (QI) becomes important and
gives rise to weak localisation (WI) which explains high values of resistivity in such materials.
Various factors as:
 thermal vibrations
 magnetic field
 spin-orbit scattering
can destroy QI and change the values of resistivity.
5
2
 1 
2
2
 1 
2
3
3
1
1
4
4
5
5
SO interaction
phonon emission
coherent quantum interference
partial decoherence
6
The MHS model and the Wigner representation
Such phenomena can be consistently described by the Morgan-Howson-Saub (MHS) model, also
known as the “2kF-scattering model”, because the contribution from backscattering
k  k
 2k F  plays the most important role in explaining the enhanced probability of return. It
is base on the quantum transport equation in the Wigner representation. We developed this
model taking into account the spin-orbit scattering.
Wigner representation of the operator  :
Aˆkk1  AˆW R, K    Aˆ
Q
K
Q
Q
,K 
2
2
expiQ  R 
(2)
where
K  12 k  k1  ,
Q  k  k1 .
(3)
Wigner function  ss R, K   density operator in Wigner representation;
s, s - spin indices.
7
Spin-orbit interaction
Non-interacting electrons moving in a static random potential V r  and a constant electric field
E experience not only an electrostatic force but also an additional force of relativistic origin: the
spin-orbit (SO) interaction. The potential which describes the SO interaction has the form
VSO r, p  aSOˆ  [V r   p]
(4)
where ̂ is the vector of the Pauli spin matrices, aSO   2mc  , and , m, c have their usual
2
meanings. SO in FZ model gives very small correction [5]. General form of the transport
equation for electrons in disordered medium with SO interaction in the Wigner representation is
2
 k

1











R
,
k

Im{
V
q
exp
i
q

r
}

R
,
k

q 



s
s
s
s
2
 m

 q
i
bSO

1





q    ss  ss R, k  12 q  

q

R
,
k

q










ss
ss
2
2 q s 
 eE  uk   R, k 
where bSO  aSO , q  V qexpiq  R  [q    2ik ] , and   f FD H  H .
8
(5)
Equation (5) is equivalent to the set of four coupled equations for
 R, k ,   R, k ,  R, k ,  R, k  .
We took some simplifying assumptions to solve this set of equations because of its complexity:
1. the diagonal elements of the Wigner matrix are identical
 R, k    R, k  ;
2. we neglect the contribution coming from the term that includes the matrix element of the
Pauli spin operator between the identical spin states;
3. we neglect the source term in equations for the off-diagonal elements of the Wigner matrix.
Due to these assumptions the initial set of four coupled equations separates into two independent
sets of similar structure. We present one of them below.
9
Decoupled set of transport equations with SO interaction
bSO
2 

 k

1
1





















R
,
k

Im
V
q
exp
i
q

R

R
,
k

q

Im

q



R
,
k

q




2
2
 
 m
 
 q 
2
 qE 
k
R, k 
m
6a 
bSO
2 

 k

1
1





















R
,
k

Im
V
q
exp
i
q

R

R
,
k

q

Im

q



R
,
k

q
0




2
2


 m
 
 q 
2

6b 
Method of solution: the Zwanzig-Mori method of projection operators (separation of the
average part of ss R, k  from its fluctuating part).
The output: independent equations for the average part of  R, k  and  R, k 

k , k  

k   0
A
A
A
k   T (1) k , k  
k   
k   qE 

k
A
k    T ( 2)

k
A

A
k    
k
1  k 
m
(7a, b)
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These equations have the form similar to the one of the Boltzmann equation but differ in
symbols. Notation:
sAs k  - the averaged part of the Wigner function;
 k  - a correction, which arises from the fact that the injection of electrons by electric field
at a specific point in space is correlated with the potential in this region;
T i  k , k  - the generalised scattering kernel that contains the fluctuating part of the inverse
operators, which act of the left-hand side of equations (6a, b).
A
A
k .
Next we concentrate on 
k  and ignore the equation for 
Justification: the electron current density is given by
j
1
k
3
  R, k    R, k 
d
R
e

 k 
m
(8)
or - using our assumption 1 - simply
j  2 e
k
k A
  k 
m
(9)
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Approximate solution
The generalised relaxation time is defined by the formula
 1 k    T 1 k , k .
(10)
k
Thus we focus our efforts on constructing approximation for T 1 k , k  . It consists of two parts
T 1 k , k   T 0  k , k   T k , k 
(11)
where the first term is the ordinary scattering part and the second one is responsible for the
spin-orbit scattering.
In the effective medium approximation (EMA) we obtain the self-
consistent equation for  1 k 


 k  k 
2
k   k  ~
k  k  1  k  k       1 

2

N
2


 1 k  
2 
2
2
  k
  2


k

k





  k  2  k 2    1 
 
 2  
 2m
 
12
(12)
Nevertheless, the transport coefficients depend on the transport relaxation time
tr1 k    T 1 k , k  1  cos kk .
(13)
k
To extract the QI effects we split T 1 into three parts as in [4]
T 1 k , k   T11 k , k   T21 k, k   T31 k, k .
(14)
The first term gives rise to the Boltzmann-like conductivity where SO correction is very small
[5], the second one is responsible for backward scattering and gives the QI contribution, while
the third one contains other multiple scattering effects and can be neglected in these
approximation. Having tr1 we can calculate the resistivity  or conductivity  :

1
m
 2  tr1 .
 ne
(15)
Conclusion
The quantum transport equations presented here give the possibility of consistent calculations of
the transport coefficients in disordered media taking into account QI and WL.
13
References
[1] Ziman J. M., Phil. Mag. 6, 1013 (1961)
[2] Faber T. E. and Ziman J. M., Phil. Mag., 11, 153 (1965)
[3] Meisel L. V. and Cote P. J., Phys. Rev. B15, 2970 (1977)
[4] Morgan G. J., Howson M. A., and Saub K., J. Phys. F: Metal Phys. 15, 2171 (1985)
[5] Spisak B. and Paja A., Acta Phys. Pol. A 96, 751 (1999)
Acknowledgements
The authors thanks the British Council and the Polish State Committee for Scientific Research
(KBN) for financial support under grants WAR/341/217 and 5 P0B 026 20.
14